The use of the term "hyperdrive" in "The Cage" was never fully explained, but the writer's guide, The Star Trek Guide (third revision, page 8) [1] suggests an origin for the term. According to the guide, "hyper-light speed" was an alternative, less preferred, term for "space warp speed". Hyperdrive later became more prominently known as the ftl drive in the Star Wars franchise. [2]

Warp drive and other faster-than-light (FTL) propulsion technologies were the linchpin of an interstellar civilization, making trade and exploration across vast interstellar distances viable. Without these technologies, these distances could not be crossed in any reasonable period of time, making interstellar civilization usually limited to a single sector. (TNG: "A Matter of Time") To put this in perspective, planets that were years away with impulse speeds could be reached in days with ships equipped with warp drive. (TOS: "Where No Man Has Gone Before")

According to "Rightful Heir", before Kahless died, he declared he would one day return, in the Borethstar system. Shortly after this, Klingons built the Boreth monastery there to await for his return. According to the episode "Day of Honor", Kahless lived in the 9th century. It is stated in Star Trek: Star Charts (p. 55) that Klingons had achieved warp capability in the year 930. Quark, however, stated in "Little Green Men" that the Ferengi would have had warp drive before the Klingons, had he delivered warp technology to Ferenginar in 1947.

Regarding Vulcan propulsion technology aboard the T'Plana-Hath, Ronald D. Moore commented, "Certainly Cochrane is credited with the invention of warp drive as we know it in Trek, so we could assume that the Vulcans were using something else – possibly a variant of the contained singularity used by the Romulans. That might have been a much more dangerous and inefficient technology which was quickly abandoned by most of the galaxy when Cochrane's system was introduced." (AOL chat,1997)

The development of the warp drive was recognized by the United Federation of Planets as the marker of an advanced society. It was only after a people developed warp drive that the Federation made contact, as codified in the Prime Directive. (TNG: "First Contact")

Despite the hardships imposed by the war's aftermath and the lack of advanced materials, Cochrane was able to build a manned warp-capable vessel using a converted Titan II missile. The successful first flight of his ship – the Phoenix – took place on April 5, 2063, and drew the attention of a Vulcan exploratory vessel, leading to the event known as First Contact. (Star Trek: First Contact)

Development of warp technology proceeded slowly over the next eighty years, after the flight of the Phoenix – due, in no small part, to the cautious advice of the Vulcans – and it was not until the 2140s that a warp engine developed by Henry Archer at the Warp Five Complex could exceed warp factor 2.

By the year 2151, warp technology was sufficiently advanced to allow a vessel to travel at warp 5, and the first Human starship, Enterprise, was built with this capability. (ENT: "Broken Bow") Although Enterprise was at first unable to fully realize this potential (maxing out at warp 4.7), the starship finally reached warp 5 on February 9, 2152. (ENT: "Fallen Hero")

By 2161, Starfleet warp drive technology had achieved the capability to reach warp 7, and these engines were being built into the latest class of Starfleet vessels. (ENT: "These Are the Voyages...")

Development and improvement of warp drive continued apace, and by the 2240s, Starfleet vessels of the Constitution-class had standard cruising speeds of warp 6 and emergency speeds as high as warp 8 (although under the right conditions, the engines could reach warp 9). These ships took advantage of a major breakthrough in warp technology that took place between 2236 and 2254, the breaking of the so-called "time barrier". (TOS: "The Cage")

At around the same time, warp engines were being redesigned to allow standard speeds of warp 8 and above. During the refit of the Constitution-class, the cylindrical-shaped nacelles were replaced with a new flattened design. (Star Trek: The Motion Picture)

According to the Star Trek Spaceflight Chronology (p. 180), the new warp drive of the Enterprise after the refit was an advanced fourth generation warp drive system known as "pulsed warp drive".

At some point in the 24th century, a new warp factor scale came into use, which placed warp 10 as a theoretical maximum. (VOY: "Threshold")

According to Star Trek: The Next Generation Technical Manual, the new scale was a more advanced function than the previous cubic power scale, with factors increasing exponentially in terms of power consumption and equivalent speed. This "new" scale also presents a maximum "cap" on the absolute velocity of a warp drive, even beyond the warp 10 limit; since energy cannot be created or destroyed, an exponential increase in power consumption would eventually outstrip the amount of available energy in the known universe.

By the time the Galaxy-class starship was being designed in the 2360s, warp technology had progressed to the point where speeds of warp 9.6 could be sustained for up to twelve hours, although warp 9.2 was considered the "red line." (TNG: "Encounter at Farpoint")

In the October 1995 issue of OMNI, science advisor Andre Bormanis stated the idea of warp factors beyond 10 in the alternative future was a recalibration of the warp scale, as ships of that era had gotten faster. Bormanis suggested the possibility that warp factor 15 was set to be the ultimate speed limit and warp 13 in that scale would have been the equivalent of warp factor 9.95 in the previous scale.

Gene Roddenberry originally intended the Enterprise to become transparent while in warp drive, as depicted in "The Cage" (later reformatted into the two-part "The Menagerie").

The idea was that the ship would be traveling faster than light, which means that light would not reach it, rendering the vessel invisible to the naked eye.
However, according to Einstein's Theory of Special Relativity, the speed of light is a constant from any frame of reference. An observer moving at close to "c" would still observe light moving toward him and away from him at "c."